Answer:
[tex]\frac{\pi}{18}(e^{10} - e^8) \approx 3324[/tex]
Step-by-step explanation:
We begin by solving for x in term of y
[tex]y = ln(3x)[/tex]
[tex]e^y = 3x[/tex]
[tex]x = \frac{e^y}{3}[/tex]
We can use the disk method to calculate the volume of rotation around the y axis
[tex]V = \int\limits^5_4 {\pi r^2} \, dy[/tex]
Where [tex]r = x = \frac{e^y}{3}[/tex]
[tex]V = \int\limits^5_4 {\pi \left(\frac{e^y}{3}\right)^2} \, dy\\V = \frac{\pi}{9}\int\limits^5_4 {e^{2y}} \, dy\\\\V = \frac{\pi}{9}\left[\frac{e^{2y}}{2}\right]^5_4\\V = \frac{\pi}{9}(e^{10}/2 - e^8/2) = \frac{\pi}{18}(e^{10} - e^8)\\ V = \frac{\pi}{18}19045.5 \approx 3324[/tex]
